2022 AMC 12B Problems/Problem 4

Problem 4

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?

$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

Solution

Using Vieta's, denote the two distinct integer roots as $r_1,r_2$ . Then $r_1 \cdot r_2 = 36$ We list out the factors of $36$ as follows: $\{1,2,3,4,6,9,12,18,36\}$ Then, we notice that there are exactly 4 pairings of distinct integers that multiply to $36$ (everything excluding $6$ ).

However, we aren't done yet! $r_1$ and $r_2$ can be negative. Since the signs of $r_1$ and $r_2$ must match and being negative doesn't change the number of options, we simply double the number of positive pairings to arrive at $\boxed{\textbf{(B)} \ 8}$ ~ $\color{magenta} zoomanTV$

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2022 AMC 12B Problems/Problem 4

Problem 4

Solution

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